Higher l^2-Betti numbers of universal quantum groups

TL;DR

This paper computes the $ ext{l}^2$-Betti numbers for universal quantum groups, providing explicit values for both the free and half-liberated cases, advancing understanding in quantum group invariants.

Contribution

It provides the first complete calculation of all $ ext{l}^2$-Betti numbers for these classes of universal quantum groups, including new results for their half-liberated versions.

Findings

All $ ext{l}^2$-Betti numbers of $ ext{U}_n^+$ are calculated.

Explicit $ ext{l}^2$-Betti numbers are obtained for $ ext{U}_n^*$.

The results enhance understanding of quantum group invariants.

Abstract

We calculate all $\ell^2$-Betti numbers of the universal discrete Kac quantum groups $\hat U_n^+$ as well as their full half-liberated counterparts $\hat U_n^*$.